General Concave Integral Control

In this paper, a class of fire-new general integral control, named general concave integral control, is proposed. It is derived by normalizing the bounded integral control action and concave function gain integrator, introducing the partial derivative of Lyapunov function into the integrator and originating a class of new strategy to transform ordinary control into general integral control. By using Lyapunov method along with LaSalle’s invariance principle, the theorem to ensure regionally as well as semi-globally asymptotic stability is established only by some bounded information. Moreover, the highlight point of this integral control strategy is that the integrator output could tend to infinity but the integral control action is finite. Therefore, a simple and ingenious method to design general integral control is founded. Simulation results showed that under the normal and perturbed cases, the optimum response in the whole domain of interest can all be achieved by a set of the same control gains, even under the case that the payload is changed abruptly.


Introduction
Integral control [1] plays an important role in control system design because it ensures asymptotic tracking and disturbance rejection.In the presence of the parametric uncertainties and unknown constant disturbances, integral control can still preserve the stability of the closedloop system and create an equilibrium point at which the tracking error is zero.The main task of the integral controller is to stabilize this point, which is challenging because it depends on uncertain parameters and unknown disturbances.

Classical Integral Control
The simplest controllers that achieve integral action are of the proportional integral derivative (PID) form that introduces integral action by integrating the error.It is well known that integral-action controllers with this class of integrator often suffer a serious loss of performance due to integrator windup, which occurs when the actuators in the control loop saturate.Actuator saturation not only deteriorates the control performance, causing large overshoot and large settling time, but can also lead to instability, since the feedback loop is broken for such saturation.To disguise this drawback, various anti-windup schemes have been proposed to deal with integrator windup or to improve transient performance.These are classified into three different approaches: 1) conditional integration and/or integrator limiting [2][3][4][5][6][7], in which the integrator value is frozen or restricted when certain conditions are verified; 2) back-calculation [8][9][10][11], in which the difference between the controller output and the actual plant input is fed back to the integrator; and 3) a nonlinear integrator [12][13][14][15][16], whose output is shaped by a nonlinear error function before it enters the controller.Some conditional integration and/or integrator limiting may not guarantee a zero steady error and could result in an oscillatory system for the step-referent input when an estimated limitation is embedded in the controller.In the back-calculation approach, the compensation for integrators is active whenever actuators are saturated; integrator windup cannot be completely avoided.For nonlinear integrators, the output still goes to infinity and integrator windup may occur.In addition, the universal integral continuous sliding mode control (CISMC) first reported by [1] has the same problem as a PID controller because it applies the same integrator.An improved version was proposed by [7], in which the integrator is modified to provide integral action only inside the boundary layer and the derivative of the error introduced into the integrator.All these integrators, except for the one proposed by [7], were designed by using the error as the indispensable element.So, all of them is called classical integral control.

General Integral Control
In 2009, general integral control, which uses all available state variables to design the integrator, is originated in [17], where presents a unified framework for general integral control, some general integrator and controller, the necessary conditions and basic principles for designing a general integrator, however, their justification was not verified by strictly mathematical analysis.In 2012, based on linear system theory, we present a systematic design method for general integral control [18] with a linear integrator on all the state of dynamics.The results, however, were local.The regionally as well as semiglobally results were proposed in [19], where presents a nonlinear integrator shaped by sliding mode manifold, and then general integral control design is achieved by sliding mode technique and linear system theory.Therein, the sprout of concave function gain integrator appeared.In 2013, based on feedback linearization technique, a class of nonlinear integrator which is shaped by diffeomorphism, and a systematic design method for general integral control are presented by [20] and the conditions to ensure regionally as well as semiglobally asymptotic stability are provided.
This paper is not a simple extension of the work [19], but it is developed as a class of fire-new general integral control, named general concave integral control in such a way of normalization.The main contributions are as follows: 1) the partial derivative of a class of general Lyapunov function is firstly introduced into the integrator design; 2) the bounded integral control action and concave function gain integrator are normalized; 3) a general strategy to transform ordinary control into general integral control is proposed; iv) by using Lyapunov method and LaSalle's invariance principle, the theorem to ensure regionally as well as semi-globally asymptotic stability is established only by some bounded information.Moreover, the highlight point of this integral control strategy is that the integrator output could tend to infinity but the integral control action is finite.Therefore, a simple and ingenious method to design general integral control is founded.
Throughout this paper, we use the notation , and that of matrix A is defined as the corresponding induced norm

  T M
The remainder of the paper is organized as follows: Section 2 describes the system under consideration, assumption, and definition.Section 3 addresses the control design.Simulation is provided in Section 4. Conclusions are presented in Section 5.

Problem Formulation
Consider the following nonlinear system, where is the controlled output, is a vector of unknown constant parameter and disturbance.The functions are continuous in In this study, the function

 
, f x w does not necessarily vanish at the origin; i.e., and v r w .We want to design a feedback control law such that that depends continuously on and satisfies the equations, so that 0 x is the desired equilibrium point and 0 is the steady-state control that is needed to maintain equilibrium at u 0 x , where y r  .For convenience, we state all definitions, assumptions and theorems for the case when the equilibrium point is at the origin of , that is, 0 .There is no loss of generality in doing so because any equilibrium point can be shifted to the origin via a change of variables.
where x g l is a positive constant.Assumption 3: Suppose that there exists a control law

 
x u x such that 0 x  is an exponentially stable equilibrium point of the system, and there exists a Lyapunov function that satisfies, for all  For the purpose of this note, we introduce the following definition and property, which is proposed by [13].
Definition 1: where  stands for the absolute value.Figure 1 depicts the region allowed for all the functions belonging to function set For instance, the hyperbolic tangent, arc tangent functions and so on.

Control Design
For achieving asymptotic regulation and disturbance rejection, we need to include "integral action" in the control law u .Thus, general integral controller are proposed as follows, where
obtain the augented system, By Assumption 1 and choosing to be nonsingular and large enough, and then set 0 and 0 x  of the Equation (11), we obtain, Therefore, we ensure that there is a unique solution 0  , and then   0 0, is a unique equilibrium point of closed-loop (11) in the control domain of interest.At the equilibrium point, y r  , irrespective of the value of w .Now, the design task is to provide the conditions on th the system e positive constants 3 c , 4 c and matrix K  such that   0 0, is an asympto all stable equilibrium point of the closed-loop system (11) in the control domain of interest, which is not a trivial task because the closed-loop system depends on the unknown vector w .This is es- tablished in the following theorem.
  0 0, clo is an exponentially stable equilibrium point of the sed-loop system (11).Moreover, if all assumptions hold globally, and then it is globally exponentially stable.
Proof: To carry out the stability analysis, we consider the following Lyapunov function candidate, Obviously, Lyapunov function candidate ( 15) is positive define.Therefore, our task is to show that its time derivative along the trajectories of the closed-loop system (11) is negative define, which is given by,   Substituting ( 12) into ( 16), we obtain, Using ( 4), ( 7), ( 8) and ( 9), we get, Using the fact that Lyapunov function candidate ( 15) is a positive define and its time derivative is a negative define function if the inequalities ( 13) and ( 14) hold, we conclude that the closed-loop system (11) is stable.In fact, means and By invoking LaSa riance p to know that the closed-loop system (11) is exponentially stable.
Corollary 1: If the function lle's inva rinciple [21], it is easy   , g x w is equal to a constant, and then the integrator can be taken as . Thus, under Assumptions 1 and 3, we only need to choose the gain matrix K  to be nonsingular and large enough such tha inequality (13) holds, and then   0 0, t the  is an exponentially stable equilibrium point of the closed-loop system (11).Moreover, if all assumptions hold globally, and then it is globally exponentially stable.Th gumen Discussion 1: compared with tegral control pro tion be e proof can follow the similar ar t and procedure.It is omitted because of the limited space.
the in posed by [19], the main differences are as follows: 1) the integral control action is not confined to the hyperbolic tangent function and can be taken as any funclonging to function set

 
, , F x   , and then the normalization of integral control action is achieved; 2) the indispensable element of integrator is not confined to sliding mode manifold and can be taken as the partial derivative of any Lyapunov function, which satisfie o th is not co atisfie s Assumption 3, and then not only the normalization of concave function gain integrator is achieved but als e partial derivative of Lyapunov function firstly is introduced into the integrator design.
3) the control element  x u x nfined to sliding control and can be taken as any control, which s  s the conditions of Assumption 3. Remark 1: The proof of Theorem 1 seems to be very simple, in fact that is not the case because there are two tedious troubles to be concealed in the stability analysis, one is that integral control action must be bounded, another is how cancel the terms on     . Therefore, for solving these two troubles above, an ingenious de as fo sign method is proposed llows: just the integrator is taken as . Thus, we not only obtain a bounded integral control action time derivative of Lyapunov function, and then Theorem 1 can be established only by some bounded information.Consequently, the ve integral control is verified.Moreover, this resulte new integrator with a concave function gain 2. This is why the control law (10) is called general concave i Remark 2: From the control law (10), vious that the highlight point of thi ontrol strategy is that the integrator output could tend to infinity but the integral control action is finite, which is the same as the one proposed by [19].This means that this kind of integral control can devote its mind to counteract the unknown constant uncertainties or disturbances and filter out the other action, and then the stability analysis is easy to be achieved in theory and actua see Figure tor saturation is easy to be eliminated in practice.
Remark 3: From the statement above, it is easy to see that: for achieving the integral control, we only need to find a control input   x u x and a Lyapunov function

 
x V x such that 0 x  is an exponentially stable equilibrium point of the system (5).Especially, when the function   , g x w condition ( 14) can be on the closed (13), is th results in a class control int , that is, m ed into general oreover, t and   x V x he mo is equal to a constant, the dilemma removed, that is, the stable conditions -loop system (11), except for the condition e same as the one of the system (5).This not only of general strategy to transform ordinary o general integral control but also the guess [17] any control laws can easily be transform integral control laws, is verified partly.M here is great freedom in the choice of such that the control engineers can st appropriate control input in on thes hand to design their own general integral controlle Based e statements above, it is not hard to know that all of them constitute a simple and ingenious method to design general integral control together.

Simulation
Consider the pendulum system [21] described by, d by the rod and the vertical axis, and T is the torque applied to the pendulum.Vie as the control input and suppose we want to regulate the an tende w T  to  .Taking 1 x     , 2 x    and u T  , the pendulum system can be written as, Now, using the linear system theory, the choice   , and then a globally exponentially stable ntroller can be given as,

Conclusions
A class of fire-new general integral control named gen-

al (solid a
eral concave integral control was proposed in this pape The main contributions are as follows: 1) the partial derivative of a class of general Lyapunov function is firstly introduced into the integrator design; 2) the bounded integral control action and concave function gain i grator are normalized; 3) a general strategy to tran ordinary control into general integral control is proposed; 4) by using Lyapunov method and LaSalle's invariance principle, the theorem to ensure regionally as well as semi-globally asymptotic stability is established only by some bounded information.Moreover, the highlight point of this integral control strategy is that the integrator output could tend to infinity but the integral control r. ntesform ac tion is finite.Therefore, a simple and ingenious method to design general integral control is founded.
In this note, only a class of general integral control was presented.It is clear that we can not expect one particular procedure to apply to all system.Therefore, new design techniques for general integral control are needed to solve the wider theoretical and practical problems.

Assumption 2 :
No loss of generality, suppose that the function   , g x w satisfies, c are all positive constants.

Theorem 1 :
Under Assumptions 1-3, if there exists a po tic y sitive define diagonal matrix K  such that the the following inequalities, which is obtained by differentiating the function     and using the partial derivative of Lyapunov function   x V x x   as the indispensable element of integrator, and then we get
k 1 and k 2 are all positive nstants.Substituting   x u x into (19) and deleting the conant term he control law in Assump n 3 can be taken st co P that s taken as V x ponentially st quilibrium point of the system (20).Therefore, f ven positive define symmetric matrix Q there exists a unique pos y n in P taki equation T PA A P Q   , and then th mption has strong robustness, fa gence and good flexibility and can effectively deal with unknown exogenous disturbances, nonlinearity and uncertainties of dynamics.

Figure 3
Figure 3. Sy turbed case (d stem output unde shed lin r norm line) and pere).